Comparison between Control and Experimental Groups

This section presents a comparison between the two groups. The data are analyzed both qualitatively and quantitatively. An independent t- test is used to investigate whether the mean scores of the two groups (control and experimental) are significantly different from each other.

4.3.3.1 Pre- control versus Pre- experimental

First, it is important to compare and note the similarity of the results between the two groups before the implementation of the unit on derivative on derivative. The results of the three parts of the questionnaires (I, II, and III) reveale d that the symbolic

representation of derivative dominated the thinking of both groups (16 students (62 %) in the control group and 15 students (58 %) in the experimental group). Both groups had a dominant "procedural" conception of derivative that consists of rules of differentiation (e.g. ( _,

, and

and others) learned by memorization. Moreover, the results revealed that both groups had deficiencies in their understanding and misconceptions on derivative; majority of students thought that the derivative of a function at a point is equal to the slope of a secant line between two points , the value of the tangent equation at a point, and the average rate of change between two points. In addition, for most students the relation between the steepness of the graph a function and the derivative concept or the

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relation between a function and its derivative were ambiguous. Finally, the independent t- test (Table 14) revealed (p = 0.59 > 0.05) that there is no significant difference in the mean scores of the quest ionnaire between the two groups before the implementation of the unit in derivatives.

Table 14

Independent t-test comparing control and experimental groups' mean scores on the questionnaire, before the implementation of the unit on derivative

Two Groups

Control Group Experimental Group

mean SD Mean SD T Sig.(p-

value)

Questionnaire 3.96 1.34 4.15 1.22 0.54 0.59

4.3.3.2 Post- control versus Post - experimental

The results of students' responses on the questionnaires revealed that there are significant differences between the groups after the implementation of the unit on derivatives.

 The symbolic representation dominates the thinking of 11 students in the control group as compared to only two students in the experimental group.

 None of the students in the control group talked about real life

applications of derivative while 10 students in the experimental group mentioned that the derivative is used in many aspects of life ( biology,

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velocity and acceleration in physics, growth population, marginal cost in economics )

 When students were asked about the meaning of derivative, only six students in the control group mentioned the three types of representation (rules of differentiation, slope of curve/ tangent line and instantaneous rate of change) in their responses as compared to 17 students in the experimental group.

 Concerning parts II and III, it is noticed that there is an increase in the number of correct answers to each item for the two groups with a higher frequency for the experimental group.

 In part III, item d, which states that the derivative of a function indicates whether the function is positive or negative, gets the lowest

response by both groups. It is suggested that the difference between the sign of a function (positive/ negative) and its variation (increase/ decrease) should be more emphasized during teaching. To many students, the two terms are the same, while in fact they are not.

 In both groups, the contradictions in students' choices decreased after the implementation of the unit on derivatives with higher frequency for the experimental group.

 Students in the experimental group outperformed students in the control group on the questionnaire. The difference in mean scores between the control group (M = 6) and experimental group (M = 9. 89) is 3.89 which is statistically significant (p = 0.00 < 0.05) as shown in Table 15.

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Independent t-test comparing control and experimental groups' mean scores on the questionnaire after the implementation of the unit on derivatives

In conclusion, based on the above results, it is clear that students in the experimental group showed better understanding that students in the control group. It seems that the concept definitions and images of many students of the control group are dominated by the procedural symbolic representation of derivative (rules of

differentiation and formal definition) before and after the implementation of the unit on derivatives. However, some students have the graphical (slope of tangent) and numerical representations (rate of change) of derivative in their minds. On the other hand, students in the experimental group have a richer concept image of the derivative concept. Their responses, which reflect their concept images, include the derivative as the slope of tangent line, the derivative as an instantaneous rate of change, the derivative as rules of differentiation and limit of quotient differences, the derivative as a measure of steepness of the graph of a function at a point. Finally, they includes the relation between and This result is interpreted by the fact that the approach used in the experimental group (Book 2, multiple representation approach visualization, Autograph) for teaching

Two Groups

Control Group Experimental Group

mean SD Mean SD df T

Sig.(p- value)

Questionnaire 6 1.56 9.98 0.95 50 -10.72 0.00

Notes. Twenty -six students on each group completed the questionnaires. The test was performed at α = 0.05 level of significance. The assumptions of the test were examined.

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derivative seems more effective than the approach used in the control group and enhances students' understanding of the concept.